题号:14621    题型:解答题    来源:2024年丘成桐大学生数学竞赛(代数与数论类)-无答案
Let $p$ be a prime number. Consider $\mathbb{Z}_p$ (resp. $\mathbb{Q}_p$ ) the ring of $p$ adic integers (resp. field of $p$-adic numbers). Clearly $\mathbb{Z} \subset \mathbb{Z}_p$.
(1) Show that the set $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, and deduce that a ma $\mathrm{p} f: \mathbb{Z} \rightarrow \mathbb{Q}_p$ can be extended to a continuous function on $\mathbb{Z}_p$ if and only if $f$ is uniformly continuous, i.e., for any $\varepsilon>0$. there exists some integer $N>0$ so that $|f(n)-f(m)| < \varepsilon$ for any integers $n, m \in \mathbb{Z}$ with $m \equiv n\left(\bmod p^N\right)$.
(2) Let $a \in \mathbb{Q}_p \backslash\{0\}$. Under what condition on $a$, the map
$$
f: \mathbb{Z} \rightarrow \mathbb{Q}_p, \quad n \mapsto a^n
$$
can be extended to a continuous function over $\mathbb{Z}_p$ ? Justify yo ur assertion.
(3) Assume that the condition in (2) is fulfilled. Can we even $\mathrm{e}$ xtend the function $f$ in (2) to a continuous map
$$
a^x: \mathbb{Q}_p \rightarrow \mathbb{Q}_p,
$$
so that $a^{x+y}=a^x a^y$ for any $x, y \in \mathbb{Q}_p$ ? Justify your assertio n.
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